Re: Guillaume's alternative definition(s) of "com"
Le samedi 16 février 2013 à 15:40 +0000, John Pryce a écrit :
> > The decoration mechanism is a combination of the following schemes for
> > inputs and output:
> > - i1: inputs have to be bounded for com to be generated
> > - i2: inputs do not
> > - o1: actual output has to be bounded for com to be generated
> > - o2: mathematical output has to be bounded for com to be generated
> >
> > If i1 is in place, the division discards com. If o1 is in place, the
> > multiplication discards com. If either is in place, the propagation rule
> > prevents the final result from being com. The motion mandates i1+o1,
> > which means that the output cannot be com. If the constraints were
> > relaxed (i2+o2), then the output would be com. Am I making sense?
>
> Very much making sense, but there's another point, which must have
> influenced me to put i1+o1 into the current definition. Namely
> assuming (as you do) the input XX=xx_dx has the standard initial
> decoration, scheme i1+o1 gives
> YY = f(XX) = 1/(2*[1,M]_com) = 1/[2,oo]_dac = [0,0.5]_dac
>
> The dac, together with the bounded input(s), suffices to prove that
> all the intermediate values, as well as the output, are mathematically
> bounded, by compactness. (A continuous function on a compact set is
> bounded.)
>
> I think this was my reason for saying that com should describe what
> actually happens at Level 2, rather than what ideally happens at Level
> 1. Namely, I claim that your i2+o2 can never produce more informative
> results than what such a compactness argument can produce using i1+o1.
> And that there is no other good reason to move to i2 and/or o2.
>
> It is VERY possible I'm wrong. Can you show that I am?
No, it seems fine. That said, it is only a reason for having o1. It is
not a reason for having i1 though and the weaker i2 would suffice, as
far as I can tell. The reason for moving to i2 is: it puts less
requirements on the inputs and thus leads to more efficient
implementations. For instance, consider the code of the addition, you
will notice that satisfying i2 is faster than satisfying i1.
Also, since dac is sufficient for my purpose, I wonder what use people
have for com. The mathematical meaning is kind of useless, since dac
encompasses it, as you explained. So it seems like the only purpose is
to actually know when a computation will give the same results
irrespective of the flavor. But what is the practical use case for that?
Best regards,
Guillaume